Deterministic Mathematical Modeling of the Platform Performances

Strojniški vestnik - Journal of Mechanical Engineering Volume(Year)No, StartPage-EndPage
UDC xxx.yyy.z
Paper received: 00.00.200x
Paper accepted: 00.00.200x
Deterministic Mathematical Modeling of Platform
Performance Degradation in Cyclic Operation Regimes
1
Nenad Kapor 1 - Momcilo Milinovic1-Olivera Jeremic1-Dalibor Petrovic2,
University of Belgrade, Faculty of Mechanical Engineering, Belgrade Serbia
2
University of Defense, Military Academy, Serbia
Abstract
The manuscript considers the modeling of extreme-capability working platforms that are
operated in periodic cycles, each cycle having a pre-defined number of operations that affect working
surfaces. A novel hypothesis is introduced about the platform-degrading effects that cause an equivalent
decrease of the successful operations after repeated cycles. Deterministic modeling, based on the basic
equations by Lanchester and Dinner, is generalized here to include coupling between parameters. The
newly developed mathematical model of performance degradation is in good agreement with both
experimental measurements and numerical simulations. It is assumed that the new variables and their
correlations link Gaussian distribution and the observed performances of the testing platforms. Relative
probability dispersions of affected surface are derived, as a new indirect referencing figure of merit, to
describe simulations and compare them to experimental test data. The model proves a hypothesis that the
degrading effects are a function of the platform capacity, frequency of operations and the number of
available cycles. Degradation effects are taken into account through an equivalent decrease of effective
operation capacities, reflected on the properties of the affected operating surfaces. The obtained
estimations of degradation could be used in planning of platform capacity as well as in the selection of
real affected surfaces in various machining systems and for a wide range of different materials.
Key words: cycles, operations, extreme machine platforms, probabilities, deterministic modeling
0 NOMENCLATURE
M p (t ) - current number of operations
m p (t ) - current number of degraded operations
Mp
mp
- operation consumptions rate
- rate the fictive changes of non-effective
M po
operations(rate of equivalent
operations)
- capacity of particular operations
m po
- initial degraded number of operations
mp(i-1) - initial degraded number from the
previous cycle
- number of cyclic operations
m p (t) - current number of operations during
i
N
S *0
Sp
Sc
i -1
degradation
- remaining un-degraded surface
- affected surface
- remaining un-degraded surface after
the (i–1)th cycle
S i (t ) - current un-degraded surface
t -
cycle period
PD * - probability dispersion of un-degraded
operations on the platform
PD - probability dispersion of working surface
in degraded platform s operations
U i - designed capability, of the platform,
 p   p - attrition rate of operations
(consumption rate of operation numbers)
– probability of each operation
t
- current relative degradation in the
pi
i cycle
(
t
)
relative efficiency of the current
ci
process
N p (t )

- operation frequency for a given
t
cycle
N p (t ) - number of operations per cycle
p
-
elementary efficiency on the surface
*
Corr. Author's Address: Name of institution, Address, City, Country, [email protected]
1
 p (t ) - current relative capacity degradation
*ci
-
relative efficiency of the current
c i -1 -
process without degradation
relative efficiency of the current
i
c
- cumulative relative efficiency of all
finished processes
*ci - cumulative relative efficiency of all
finished processes without degradation
process, after i-1 cycles with
degradation effect
1 INTRODUCTION
Machines operating in cycles and their
properties are not studied in depth in literature,
and as such are not well described by integral
mathematical models. If the effects of their
operation are actions on the given working
surfaces under given constraints, then the quality
of the affected surfaces can be described by
reliability functions. In this way the operating
capabilities of the platform can be determined. A
majority of the published papers use standard
aproach to the measured performances that depend
on the machine's designed purposes. Such
processes are described in [1] – [3] for the
Abrasive Flow Machines (AFM) where material is
hardened by randomly treating the working
surface with abrasive particles with polymeric
fillers and dispersed within the flow media. The
authors of [1] classified the work piece parameters
into three groups based, among others, on the
number of cycles (operations), and the machining
time. Some of these parameters were determined
experimentally in [2], where the authors
recognized that the parameters denoted as the
creeping time and the cycles frequency have
impact on the quality of the machining process. In
[3] the authors experimentally prove that the
mentioned parameters influence the process.
Common for all three papers is that they do not
include hidden random effects caused by particles
affecting the surfaces in cyclic operations,
although such effects significantly influence the
quality of surface treatment. In all three papers the
mathematical modelling of the process is missing.
Another similar type of machines with
cyclic operation affecting working surfaces are
those described in [4] as the shot peening (SP)
platforms. They bombard a surface with spherical
beads to increase the material fatigue strength. The
physical modeling of the influence of the bead
shapes on the performance of the surface
hardening processs is presented in [5]. Random
surface effects due to bombing cycles are a result
2
of the quallity of the machine performance.
However, the connection between the effects and
the particular operations is missing in [5]. Paper
[6] utilizes a risk function to consider the example
of sun-rays hitting a surface as a random process.
In fact, the determination of the risk function
dumping requires much more precise estimations
of probabilities distribution of the effect occurance
on the attacked surface. However, the mentioned
paper is missing the mathematical model of
random disturbances of these probabilities.
Common for the description of the processes in
both AFM and SP machines, as well as the
processes described in [6], is that they lack the
deterministic or probabilistic mathematical
modeling of cycles and their parameters on the
final process efficiency.
Mechanical engineering of extreme
machines in defence technologies have particular
operations grouped in the cycle regimes. These
operations affect the working surfaces or areas,
with constrained machine capacities with regard to
operation numbers. The modeling of efficiencies
in such cases is usually done using the
deterministic diferential theories of operational
research. This approach is based on the so-called
Lancester and/or Dinner equations of particular
probabilities and their distribution laws, as
presented in detail in [7], as well as in [8]. Their
equations use variable attrition rates as the
frequencies in operations probabilities, similar to
[9], where surface point effects are taken with
variable probabilities. Modeling of cycles
efficiencies in these references is done by coupled
equations, where two subjects simultaneously
affect each other. Their actions are interdependent,
but diferent. Their efficiencies also differently
evolve with time. This approach is not fully useful
to define a stand-alone efficiency estimation for a
single subject.
A mathematical model of the equipment
with constrained capacities which generates
identical repeated operations in a given order is
presented in [10] for air platform equipment. The
main contribution of that paper is the treatment of
the action on the working surfaces as a random
process, but the probability distribution laws on
the affected surfaces are missing. The twodimensional Gaussian distribution laws, used for
welding processes as referred in [11] could be
useful in the estimation of random processes on
surfaces.
According to the state-of-the-art analysis as
presented in the quoted papers, there is no
comprehensive mathematical model, proven by
experimental data, that would be capable of
explaining relations between the machine's cyclic
performances, its capacities and the quality of
randomly affected surfaces, as well as the
designed processing time. This is because the
mentioned papers do not consider two repeating
processes simultaneously acting on a single object,
one as working and second as redundant or
parasitic, which together changes the quality of the
expected performances.
The objective of this paper was to develop a
general joint mathematical model that includes all
pertinent factor that influence the final efficiency
of procesing, thus enabling simulation and
evaluation of these parallel processes.
Based on the specific requirements for
sequential processing of the surfaces, a
mathematical model is developed using a
deterministic approach treating the surface
processes as random variables.
The objective was to test the efficiency of
cyclic operations affecting the working surfaces,
basically by considering differences caused by the
capacities and operation rates of the processing
machine. This was shown using the experimental
data on operation platforms with extreme
performances.
The novelty of the aproach presented in this
manuscript is the redesign of coupled
deterministic equations done in a new manner. In
the mentioned literature these equations are
employed to describe the mutual effects of objects
as a function of the elapsed process time. This
approach in literature makes the time functions
dependent on the performances of two objects. In
our approach one object executes two operations
in parallel, one of them comprising the working
process itself, and the other, parasitic, occuring as
self-degradation dependent on the first one. Both
happen on the same object -- the operation
platform performing the same action. The new
aproach composes deterministic equations to
describe this and to measure changes of the
platform efficiency. By our approach the quality
of the working process is the convolution of both
kinds of operations in one cycle, as well as their
frequencies. The number of cycles influences the
random arguments and reflects on probabilities of
working surface coverage that obeys the twodimensional Gaussian distribution laws. This was
taken as the measure of the changes of quality due
to self degradation.
2 EXPLANATION OF THE GENERAL
MODEL
The model offers the possibility to evaluate
the degradation of the platform performance,
regarding the equipment and devices contained
within the platform. The quality of the affected
surface is regarded as the dimension of probability
dispersion. This dimension appears during the
execution as the consequence of cycle duration
and the operations frequency, as well as of the
capacities of the platform. The approach presented
in [12] which developed operations frequency
coupled with execution probabilities as the
combined attrition parameter was used in
developing our general model. Changes of
probability dispersions of random values on the
affected surface appear in the form of Gaussian
distribution law. The degradation of the platform
properties through the operation cycles is
represented by changes in the Gaussian
distribution. This is valid under assumption that
for each cycle in the working regime one
particular Gaussian distribution function is valid.
In our approach this function is distributed
in successive cycles in the form of extended
probability dispersions of both random arguments
in the two surface directions. Consequently, this
means that the changing of efficiencies over time
is measured by the resulting effects on the new
randomly affected surfaces. The decrease of the
efficiency with each new cycle reflects on the new
less affected surfaces. This also means the
degradation of the working platforms capabilities
caused by less effective particular operations in
the cycles. The cause of this degradation could be
a consequence of rapid high energy operations
realized in short-time sequences (high mechanical
power values) in successive, orderly repeated
cycles, similar to those described in paper [13].
But, in the mentioned paper, the affected points on
3
the surfaces do not obey any probabilistic law, and
thus there is no error distribution as a modeling
parameter. In our research we use the changes of
the probability dispersion (PD) after each cycle
due to all errors in the cycle as a measure of the
platform efficiency. These changes are caused by
the generator of the cycles and by its selfdegradation and are reflected in the decreased
number of declared operations. This makes the
designed operational capacities of the platforms
less efective with the number of cycles.
In order to estimate the degraded platform
performances by means of time-based simulation,
new relative parameters have been accepted in the
modeling.
The deterministic modeling of the
estimations of the so-called vulnerability
performances is presented in [14] and [15]. The
performances considered there are similar to our
degrading platforms performances. The models
presented in [16], called PEXPOT, LEVPOT,
DYNPOT, were also developed as vulnerability
considerations based on attrition rate function and
thus indirectly describe the kind of expected
degradation capabilities. An esssential diferrence
of our model is that the degradation of the system
appears as a direct consequence of selfdegradation caused by the effects of the repeated
cycles. The designed frequencies and functional
probabilities, contained in each operation, is
reflected through the full platform capacity on the
affected surfaces. This effect makes the proposed
model more useful in planning the redesigning of
platform capacities for required affected surfaces.
3 MATHEMATICAL MODELS
In the presented model the platform has the
capacity of M p0 particular operations oriented
toward the working surface. These operations
occur in dynamic regimes with successive
frequencies  and probability of surface action
of about p = 0.997. This is provided using the
maximum technical dimensions of the surface,
which correspond to the 64 PDav2.Width and
depth of the surface used eight same average
probability dispersions PDav, in both surface
directions. Average probability dispersions PDav
is taken as an equal of the expected Gaussian
distribution
of
two-dimensional
random
arguments. The probability variations are
represented as functions of the cycle number and
4
of the full capacity of operations. The designed
properties of these processes are consequently the
function of probability changes. The adopted
hypothesis is that the degradation of the platform
performances is an imaginary effect, able to be
explained by the values of effective and
ineffective number of operations. This ensures a
possibility to consider the ineffective number as a
value increasing with the number of cycles during
the exploitation time. In that sense the increasing
number of ineffective operations corresponds to
the increase of cycle probabilities dispersions.
Operative consumption is realized in cycles
with the same sequential probabilities of
operations, p, like in [10]. In that case, the
frequency of executions of real operations, as the
real rate of operation is
dM p
  p   p .
dt
(1)
This determines the remaining number of
operations as the M p  M p (t ) in each moment
of time t in the cycle duration interval.
It is expected that probability would not have
a fixed value but will vary over exploitation time.
The changes of probability function p could mean
random changeable performances which disturb
the rate of real operations on the working
surfaces. The probabilities changes affect the rate
of real operations M p in Eq. (1). This is not
really possible because the frequency of operation
executions is a designed property of the platform
hardware. The present hypothesis has only an
imaginary effect. The acceptable solution could
be to recalculate the influence of the number of
ineffective operations on the new probable
dispersion PD reflected in a new Gaussian
distribution but for the unchanged execution
operations probabilities. The consequence is that
the model has to consider the extended working
surfaces, with new dimensions 64 PD2 engaged in
operations after each cycle.
The platform degradation, as an imaginary
effect, is a process in the real cycle time and
simultaneously parasitic in real operations. A new
value of the modified equivalent number of
operations m p (t ) is diminished by this imaginary
effect. This is generated as a current and
recalculated capability of working platform. The
new value is lower than the real number of the
remaining
operations M p (t ) .
At
the
very
beginning it is equal to the real available capacity
m p0  M p0 .
The reason is underpinned by the fact that
the model of self-degradation is viewed as a new,
fictive rate of equivalent non-effective operations
changes m p , which is not equal to the rate of real
operation M p . This orients the mathematical
model to consider the share of degraded value on
each of the real operations, and by that effect,
redesign the remaining number of operations
available on the platform. Such transformation
implies that the degrading rate of m p and the
real rate M p during each cycle are proportional
to
the
number
remaining
of
equivalent dimensionless
1
operations
. The correction
mp
equal time intervals for each cycle of t , and the
rate of operations in the cycle is the same, then
using (3), any of i cycles (where i = 1,2,……n) is
used at the beginning of a new, redesigned
equivalent number of operations from the
previous cycle. The current relative degrading in
cycle, is defined as a new differential equation
dp
1
i
, i = 1,2,3.
  p
dt
 p m2p
i
 i -1
(4)
The solution of (4) is
 p  t   1i
capacity, the platform performances are degraded
continually with each cycle. This means always a
new valid number of operations with regard to the
remaining capacity. It is inappropriate to use the
approach as constant and fixed for any platform
capacity since it is dependent on the available
number of operations. The relative degradation of
the platform capacity is taken as more acceptable
in modeling with the functional ratio
 p (t )  m p (t ) / m p where  p (t  0)  1 , as the
0
current relative capacity degradation of the
platform. General differential equation (2) of new
functional  p , by methodology given in [7,], is:
dp
dt
  p
1
 p M 2p
0
(3)
If the platform, under the same conditions,
executes repeating working cycles n times, in
2
t.
mp
(i-1)
(5)
The function of the current relative capacity
degradation of the platform full capacity after
(i–1), and during the i–th cycle at an instant
i -1 t t i t , similarly to [10], is
i-1
p  t   p
 t   p
i
coefficient is the portion of one operation within
the actual remaining equivalent number m p .
Based on the previous concept, the
differential equation for the degradation rates, Eq.
(1), becomes
dm p
1
  p
dt
mp
(2)
Since the model of equivalent numbers is a
function of time and the current equivalent
number m p , as the instantaneous remaining
2 p
j1



j
(6)
i-1
with the condition   p  1 for the i  1 . The
j 1
 j
estimation of the relative efficiency of the current
process is the function of the affected and the
initial working surface. This functional is
determined for the unaffected, remaining surface
at each cycle and the final working surface the
from previous cycle, taken as initial in the current
one. It is given in the form
S i (t )
ci (t ) 1- c .
S i -1
(7)
Differential equation of the relative efficiency
of the current process, as the remained relative
surface within operation cycle considered as the
degraded ones, according to a similar differential
equation in [7,10], is
d ci
0 t
t
-U i pi ci ,
(8)
dt
If the platform operates in cycles without
1 , the functional
degradation effect, its pi
 p does not affect Eq. (8) and the coupling of
i
Eq. (4) and Eq. (8) is lost. Consequently, the
relative efficiency of the current process, denoted
5
by *ci , during the un-degrading surface
processing in the cycles is described by
d *ci
-U i *ci .
(9)
dt
In both (8) and (9), the operation number in
one cycle is the designed capability, and could be
variable. This depends of the designed cartridge
capacity used for continual operations in the short
impulse regimes. The well balanced example
between the number of operations and the covered
affected surface in one cycle is the referent
platform given in [10]. It uses cartridges of
maximum N p ( t ) 8, and its cycle expires in
4.4 seconds. The accepted functional designed
capability, of the platform, redesigned for the
considered example is:
1.82
Ui
p
So
(10)
Appropriate values need to be calculated for each
platform
cartridge
with
their
declared
performances regarding the expected affected
surface. The solution of Eq. (8) is:
ci
t
ci
U im2
p i -1
3 p
0e
3
pi -1
,
(11)
while for the ideal, un-degraded effect, from Eq
(9), it takes the form:
*ci t
*ci 0 e
U it
.
(12)
The next appropriate assumptions for the initial
conditions are used:
A. Model with initial conditions for the relative
efficiency of the current process at the very
S (0)
beginning ci (0) 1- ic
1.
S i -1
B. Model with variable relative efficiency of the
current process at the very beginning
0
1 taken in the next cycle
ci t
ci i -1
from the end of the previous one. In both cases,
the designed cycle capability of the platform
1.82
U i max
p
const and is constant in all
So
cycles as a declared value.
Cumulative relative efficiency of all finished
processes is
6
n
c   c
i 1
i
(13)
For the experimental verification of the
correlation between the platform capacities and
the affected surfaces for the degraded as well as
for the un-degraded (available) number of
operations, the new expressions were required. If
all available cycles on the platform expired, the
full affected surface Sp and the working surface S0
can be correlated. The correlation could be
expressed by a relation analogous to (7), using
degradation effects on the surface given by
cumulative relative efficiency of all finished
processes in (13). This yields the relation
S0 
Sp
1  c
.
(14)
The same logic, analogous to expression (14),
could be used for the un-degraded working surface
S*0 and the affected surface S*p equations. They
also have to be related with the cumulative relative
efficiency of all finished processes without
degradation of  *c , which is a product similar to
(13), with  * determined from (12), in the
c(i)
same form as (14). For the ideal designed number
of operations, as well as for the degraded number,
the particular affected surfaces are equal (S*p 
Sp). This is because no operation in the cycle is
missed, but is only disposed somewhere on the
larger area. The working surfaces So and S*o could
be determined as the squares of the appropriate
Gaussian linear values of average probability
dispersion PDav for both cases. Since the degraded
and the un-degraded cumulative relative efficiency
of all finished processes are the functional
correlated to the surfaces given by (14), the
dispersion values, in both cases, also satisfy these
correlations.
Since the surfaces are taken as 8PDav x 8PDav,
using basic Eq. (14) and their analogues, the ratio
of PD* in un-degraded and PD of degraded cases
is expressed by
PD *
PD

1  c
1   *c
.
(15)
This new approach provides the method for
comparison of degraded and un-degraded surfaces
of accepted dimensions, or to treat expected
degrading by constraining the allowable ratio.
4 SIMULATION DATA AND RESULTS
decreasing number of equivalent operations after
each real execution rate. The most rapid decrease
of the current relative capacity degrading has
been observed for the platform with 8 designed
operations within one cycle.
1
[o]
0,98
0,96
0,94
8 Mp
16 Mp
32 Mp
40 Mp
24 Mp
0,92
pi
Simulation tests are realized using MATLAB
software package and compared with experimental
research. The basic assumptions in the simulations
were
- Platform capacities and cycles performances
presented in Tab. T.1., are used for simulation
testing, as well as in experimental modeling.
These data are used in the platform degradation
modeling
- Numerical test is provided for the 8, 16, 24,
32, and 40 operations capacity of the platform. As
it was accepted in the mathematical model, one
cycle had 8 operations and expired in a nominal
time of 4.4 seconds.
Figure.1 represents the simulation results of
the current relative degradation of successive
ordering operations in the sequential cycles. The
degraded values are positive and decreasing,
bounded with value 1 from the upper side. The
simulation shows that at the end of each cycle, the
current relative degradation final value is
decreasing, regardless of the number of operations
remaining constant.
0,9
0,88
0,86
0,84
0
10
20
30
40
50
60
70
80
90
100
[s]
t
Fig. 1. Functions of current relative degradation
of successive ordering operations in the
sequential cycles
Lower degradation occures on platforms
that have 16, 24, 32 and 40 operations in each
cycle.
Platforms with higher initial capacities, like
e.g the number of operations, have smaller
gradients and lower degradation at the end of the
cycle.
40
35
30
Table 1. Simulated platforms performances
Capacity variant
Number of operations in one tool set
The number of tool-sets
Available Number of platform s
operations
Rate of operation  [1/s]
Working surface [m2]
Effective tool radii Rc [m]
Elementary surface effciency [m2]
Number of designed affected
operations in one cycle
mp
25
1
2
3
4
5
4
4
4
4
5
2
4
6
8
8
8
16
24
32
40
1.8
1.8
1.8
1.8
1.8
1.7
1.7
1.7
1.7
1.7
0.126
0.126
0.126
0.126
0.126
0.05
0.05
0.05
0.05
0.05
8
8
8
8
8
20
15
The consequence of the model is that
platform capability degrades more rapidly in each
subsequent cycle, regardless of the same number
of operations. It is the consequence of the
8 Mp
16 Mp
32 Mp
40 Mp
24 Mp
10
5
0
0
10
20
30
40
50
t
60
70
80
90
[s]
100
Fig. 2. Real initial and degraded equivalent
number of operations on the platforms with
different operation capacities
Figure 2. represents the equivalent number of
degraded operations starting with a different real
number of platform capacities. It is visible that the
gradient disperses with time. This process shows
the minimum gradient for the platforms tested
with 40 or more real operations compared to
platforms with less capacities used in the
simulation model.
7
c
0,7
0,6
0,5
0,4
0,3
0
5
10
15
20
[s]
t
o
[]
1
INITIAL CONDITIONS
B model degrade 40 operation capacity
B model un-degraded 40 operation capacity
B model degrade 32 operation capacity
B model un-degraded 32 operation capacity
B model degrade 24 operation capacity
B model un-degraded 24 operation capacity
B model degrade 16 operation capacity
B model un-degraded 16 operation capacity
B model degrade 8 operation capacity
B model un-degraded 8 operation capacity
0,9
0,8
0,7
c
0,6
0,5
0,4
0,3
0,2
0,1
0
0
5
10
15
t
20
[s]
Fig.3 Current relative efficiency of affected surface
with variable operation designed capacities and
number of cycles a) model A and
b) model B
Also paper [1] reported significant changes
in the first 20 operations (so-called cycles in [1]
and [2]) and reaching of a saturated level
gradually. The result shown in Fig 1. also shows
saturation performance of relative degrading , but
with at least 40 operations. It could be seen that
those values are of the same order of magnitude.
Experiments in [1], developed for the
surface roughness, have shown that after 50
operating cycles the roughness decreased slightly.
Comparative considerations about similarities of
surfaces affecting processes, shown in fig 3
regardless that they have been tested on the
8
1
2
3
4
5
6
PDav
7.5
5.0
5.0
5.0
5.0
5.5
5.5
36
10
10
10
10
0
12.67
5.0
5.0
9.0
9.0
9.0
7.5
7.4
Dispersion
error [%],
model A
0,8
Model A
Experimental
of PDav data
[m]
INITIAL CONDITIONS
A model degrade 40 operations capacity
A model un-degraded 40 operations capacity
A model degrade 32 operations capacity
A model degrade 24 operations capacity
A model degrade 16 operations capacity
A model degrade 8 operations capacity
0,9
Dispersion
error [%],
model B
1
ModelB
Experimental
test of PDav
data [m]
[o]
different machines with different purposes,
showed the same sort of behavior.
Two models of surfaces with different initial
conditions marked as A) and B) shown in figure 3
are tested in simulation. The first tested case ,for
the both models, is degraded platform
performance and the second one is the undegraded performance. Both curves of the cases,
for both models, are shown in figs 3a and 3b. The
platforms with lower operation capacities, 8 and
16, in simulations have greater differences for
degraded and un-degraded tests for the model A.
This is also valid in the model B. At higher
operation capacities, 24, 32, 40 have diminishing
differences except in the model B where their
differences disappear more rapidly than in the
model A. The curves from figure 3 present relative
efficiency of the current process, concerning cycle
time as a continuously varying value.
Differences between relative values of
affected surfaces in the degraded and un-degraded
cases have been recalculated on their new
extended working surfaces dimensions and
transformed into their probability dispersions
ratios. Average probability dispersion, PDav, as the
measure of degraded surface is shown in
experimental and simulation test data in table
T.2.a and T.2.b from Eq. (15) for the platforms
with different capacities and cycle numbers
composed in the tool sets (table T.1). Repeated
experimental tests have been realized in cycles
with 8 operations in 6 realized experiments shown
in T.2a. It is obvious for both experimental
models, B and A, that they show strong
degradation of the initial probability dispersions.
The decrease is slowed down by the increasing
number of operations in the new cycles. Saturation
is achieved after 24 operations for the model A
and after 16 operations for the model B.
Table 2.a Experimental data for the probability
dispersions
Cycle No
Figure 3 explains the model of cumulative
relative efficiency of all finished processes. As it
is shown in figures a and b, the unaffected surface
decreases with the increasing number of
operations. This decrease is more or less similar to
the experimental results of surface roughness
decrease, presented experimentally in [1]. This
paper has shown similar surface effects as our
model.
32
32
22
22
22
1
21.8
Simulation of the probability
dispersions
Degraded
Total cycles
time t [s]
c  1
4.47
8.94
13.41
17.88
22.35
c  1
4.47
8.94
13.41
17.88
22.35
Un-degraded

 c [0]
1   c [0]
* 0
c[ ]
1 
* 0
c[ ]
PDav *
PDav
M
O
D
E
L
''B''
0.936
0.468141
0.117549
0.012616
0.000476
0.769
0.394497
0.100695936
0.010803896
0.000401542
0.064
0.531859
0.882451
0.987384
0.999524
0.231
0.605503
0.899304
0.989196
0.999598
0.526361
0.937217
0.990586
0.999084
0.999963
47.36
8.675
2.84
1.217
0.608
M
O
D
E
L
''A''
0.936
0.572717
0.288294
0.120871
0.041622
0.769
0.482932
0.247261184
0.103355175
0.035244115
0.064
0.427283
0.711706
0.879129
0.958378
0.231
0.517068
0.752739
0.896645
0.964756
0.526361
0.909042
0.972362
0.990184
0.996689
47.36
10.67
4.36
2.4
1.556
In the table T.2.b dispersion in the model B
is saturated after 24 operations from 47% to 2.84
% and further approaches 1%. In both models,
after at least 40 operations, (in experiments after
48), the probability dispersion PD stopped
decreasing and diminished.
The percentage of the errors measured by the
surface extension over PD, for the model in
simulations and in experiments is about 4 % for
the model B and about 29% for the model A.
This is because, in the case B, the decreased
unaffected surface from the previous cycle is taken
repeatedly as an initial surface for each next cycle.
Consequently, errors increase as the cumulative
values from the previous cycles. Taking into
account the unsteady-state random processes on
the designed platforms, predictions of efficiency
made by this indirect modeling are sufficiently
good.
5 CONCLUSIONS
In the present state of the art, in the field of
surface affecting machines, there is no unified
model theory that connects the variable machine
Operation
capacity
TableT.2.b
performances with effects on the affected surface
determined in continuous working process. The
operational research modeling offers some models
of interaction of two interdependent objects. In our
model one object, a surface affecting machine as a
platform operating in cycles is self-degrading due
to the effects of sequentially repeated cycles. In
our hypothesis it was assumed that the rate of
degradation is variable and proportional to the rate
of real operations divided by the actual remaining
% of surface
extension
It is obvious that model B where the
relative efficiency of the current process at the
very beginning of cycles is taken from the
previous cycle, has smaller saturation decrease of
PDav in both experimental and simulation cases.
8
16
24
32
40
8
16
24
32
40
capacity. This novel approach is proved indirectly
for the effects occurring on the working surface.
These effects are measured by determining the
statistical variations of the surface parameters.
The relevance of this model is that the real
effects by the cyclic operations and also by the
planned operation capacities could be predicted
for the required surface effects. The model is also
able to predict the extreme machines performances
based on the expected quality for the surfaces
treatment.
The presented simulation results are consistent
with the experimental data and coincide with other
researches referred.
Further research in this area could be the
extension of the experimental data base in order to
improve the simulation model according to the
type of special extreme machines and their
affected surfaces.
The model can be readily applied to additional
processes and effects. This could be done by
exponential redesign of the basic correlations
between the rates of operations and the rates of
degradations, or by including variable frequencies
and probabilities.
9
6 ACKNOWLEDGMENTS
This paper is a part of the research within the
Project III 47029 for MNES-RS in 2014.
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